A Geometric Perspective on First-Passage Competition

TL;DR

This paper investigates the large-scale geometry of first-passage competition on integer lattices, introducing a new construction, analyzing deterministic analogs, and identifying critical phenomena, thereby extending previous results in the field.

Contribution

It introduces a new, simple construction for first-passage competition applicable to arbitrary disjoint sets and defines a deterministic Euclidean model, linking stochastic and deterministic growth.

Findings

Large deviations estimates show the random process approximates the deterministic one on large scales.

Identifies critical phenomena when one species occupies an entire cone exterior and the other a single interior site.

Extends previous work from half-spaces to general cones and strengthens earlier finite configuration results.

Abstract

We study the macroscopic geometry of first-passage competition on the integer lattice $Z^d$, with a particular interest in describing the behavior when one species initially occupies the exterior of a cone. First-passage competition is a stochastic process modeling two infections spreading outward from initially occupied disjoint subsets of $Z^d$. Each infecting species transmits its infection at random times from previously infected sites to neighboring uninfected sites. The infection times are governed by species-specific probability distributions, and every vertex of $Z^d$ remains permanently infected by whichever species infects it first. We introduce a new, simple construction of first-passage competition that works for an arbitrary pair of disjoint starting sets in $Z^d$, and we analogously define a deterministic first-passage competition process in the Euclidean space $R^d$, providing a formal definition for a model of crystal growth that has previously been studied computationally. We then prove large deviations estimates for the random $Z^d$-process, showing that on large scales it is well-approximated by the deterministic $R^d$-process, with high probability. Analyzing the geometry of the deterministic process allows us to identify critical phenomena in the random process when one of the two species initially occupies the entire exterior of a cone and the other species initially occupies a single interior site. Our results generalize those in a 2007 paper of Deijfen and H\"aggstr\"om, who considered the case where the cone is a half-space. Moreover, we use our results about competition in cones to strengthen a 2000 result of H\"aggstr\"om and Pemantle about competition from finite starting configurations.